JSH: Why of prime gaps
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From: Mark Murray on 18 Jul 2010 06:44 On 18/07/2010 11:08, Penny Hassett wrote: > Would you copy his work verbatim or would you try to interpret what he > wrote? I don't know which faults you are trying to compensate for but I > observe that he put up what he said was a provably NP solution to the > traveling salesman problem but when I asked him for a proof he fobbed me > off with a comment that it was obvious if one looked at the phase space, > whatever that meant. Having a wiki isn't going to make his writing any > clearer but will make it harder to rewrite history. My approach would to be as fair as possible; quote verbatim, keep up-to-date, note contradictions and follow up on corrections and retractions. Clarifications by JSH and/or other parties could/should be edited into a summary. Keep references to originals too. For the sake of not rehashing old arguments, a record of associated rebuttals should also be kept. These are notably absent is JSH's own fora. Something like that. It can evolve as necessary. But This is really MichaelW's baby, so he's in the driving seat :-). M -- Mark "No Nickname" Murray Notable nebbish, extreme generalist.
From: JSH on 18 Jul 2010 11:15 On Jul 17, 10:32 pm, David Bernier <david...(a)videotron.ca> wrote: > JSH wrote: > > [...] > > > > > > > An example of twin primes is: 11, 13 > > > The gap between them is exactly 2, and one way of looking at "why" is > > to note that if for any odd prime p less than sqrt(11), if (11+2) mod > > p is 0, then the prime gap can't occur. May seem trivial but it is > > the key to understanding the twin prime gap. > > > For instance look at 13, where the next prime is 17. That's because > > (13+2) mod 3 = 0. > > > That's it. It's the only reason. Mathematically THERE IS NO OTHER.. > > > But also notice it means that 13 mod 3 = 1 is needed. So if 13 didn't > > allow itself to have 1 as a residue modulo lesser primes then that > > would have been a second twin primes case, but that is ludicrous. How > > could the prime 13 decide that it doesn't like a particular residue > > mod 3? > > > And that's your clue. If you understand that one thing then you can > > grasp the why of twin primes and then of arbitrary even prime gaps, as > > for instance for that gap of 4 between 13 and 17, you needed (13+4) > > mod p, where p is an odd prime less than sqrt(13) to not be 0, for, > > once again 3. > > > Mathematics doesn't need anything else to say a prime gap is there! > > There is ONLY one way to get a prime gap, which is for > > > (p_1 + g) mod p_2 > > > to NOT be 0 for any odd primes less than sqrt(p_1). If it is, then > > that gap does NOT occur. > > > So the requirement is that p_1 NOT equal -g mod p_2, for all primes > > p_2 less than sqrt(p_1). > > [...] > > Sorry. I have a proof secured in a safe that, for some large > constant C > 0 and for the case g = 2: > > "For every prime p_1 > C, there exists a prime p_2 with > p_1 == -2 (mod p_2) and with p_2 < sqrt(p_1)." An assertion of prime preference--it's wrapped up in your actual statement! So your assertion is that at some critical level which you call C, the primes above that level so dislike -2 as a residue that they avoid it COMPLETELY and PERFECTLY out to infinity. I humanize the primes so that the choice aspect of your assertion is clear. Why would THOSE primes decide that they are unlike their brethren at lesser values and no longer can stand -2 as a residue? Did they tell you in any messages from that great level? Possibly a spirit transmission to you in a dream? James Harris
From: JSH on 18 Jul 2010 11:28 On Jul 18, 3:08 am, Penny Hassett <Penny_Hass...(a)invalid.invalid> wrote: > Mark Murray wrote: > > On 18/07/2010 08:07, Penny Hassett wrote: > >> *There was a proposal for a JSH Wiki a while ago which I didn't support > >> because I think it should be started by you. Blogs are all very well but > >> going to the front page of yours didn't find the definition I wanted and > >> I had to use a search engine. I've said before that I wish you would do > >> something like that so that I could know which of your proposals you > >> still support. For example, in the past you have said that > > >> a) the integers extended by sqrt(2) gives the rational numbers > >> b) the rational numbers extended by pi gives the real numbers > > >> and I don't know your current thinking on these. > > > I doubt he does either. > > > The closest you'll get is his Scribd pages at > >http://www.scribd.com/jstevhbut this doesn't cover the two specific > > points above. > > > As you have noted, James is very unclear in his writing, and is on > > record as not being able to follow some of his own work. He edits > > chaotically, so it is often not clear what he currently still believes > > during the course of a thread, and later as he revisits the subject. > > > Having someone other than JSH put up a Wiki/Blog would allow a modicum > > of clarity into the collection by at least attempting to correct the > > above faults. It would also allow a separation of the mathematics from > > the hubris. > > > M > > Would you copy his work verbatim or would you try to interpret what he > wrote? I don't know which faults you are trying to compensate for but I > observe that he put up what he said was a provably NP solution to the > traveling salesman problem but when I asked him for a proof he fobbed me > off with a comment that it was obvious if one looked at the phase space, > whatever that meant. Having a wiki isn't going to make his writing any > clearer but will make it harder to rewrite history. Hey, I'm replying first here but will go back to your previous post to see if I can reply there as well. And I welcome the chance to talk to you again! I value your postings and am sorry that my attempt at explanation with my TSP idea didn't satisfy you. But that was comments on my math blog. Not exactly the best space. Why not join my math group? There are some posters here who don't like me who are on that one and "Mark Murray" was on for a while until I had to ban him though he was posting under a pseudonym it wasn't hard to figure out it was him. Oh, so why was he banned? He started posting some wacky assertions that I was blocking some of his posts when I didn't. I warned him that I didn't have to tolerate a running argument of false assertions on that group and that he'd been banned if he didn't stop. He didn't stop so I banned him. I run my math group loosely but if you get wacky on it, it's not sci.math so yes I can stop an argument there. And I did not block any of his postings (until the ban of course blew them all off). Why bother? It was all rather odd actually and I digress, but it seems weird to me. In any event I welcome the opportunity to discuss math issue with you, and oh yeah, as for my TSP idea, one of the posters on my math group who posts here as well is "rotwang" and he has asserted counter-examples to it. I've backed down to saying I like the approach and am not currently defending it as a solution to TSP, nor am I working on it further at this time. So it is on a back-burner. But also I've noted I'm curious about Google search results that pull it up at #2 or #3 behind Djiskstra, with the search: optimal path algorithm I DO rely on multiple inputs to see if there are interests in my work, so notice when some of my research percolates up by Google search results. At this point in time I have so many research ideas that with such a large body of work it is difficult for me to keep up with them all on my own, or to have a good handle on which particular idea may be drawing interest at any particular time, so search engines help out in that way tremendously. James Harris
From: Nell Fenwick on 18 Jul 2010 12:06 "MichaelW" <msjmb(a)tpg.com.au> wrote in message news:4c427a83$1(a)dnews.tpgi.com.au... > On Sat, 17 Jul 2010 20:14:51 -0700, JSH wrote: > >> >> Another way of looking at it is, if you're looking at bigger and bigger >> primes p_1, and you are getting all these residues modulo primes less >> than sqrt(p_1), and there is no preference by the primes then just at >> random at times you will have cases where -g is not a residue modulo ANY >> of those primes which will give you a prime gap. >> >> James Harris > > The only part of the post I would take issue with is with this paragraph. > > Let's see if I get this right. For any given prime p_1 we can state the > probability you said "probability", which is way outside JSH skill set.
From: Joshua Cranmer on 18 Jul 2010 14:16
On 07/18/2010 11:15 AM, JSH wrote: > Why would THOSE primes decide that they are unlike their brethren at > lesser values and no longer can stand -2 as a residue? Have you ever heard of "Strong's Law of Small Numbers"? It is a law which notes that there exist many things which appear to be patterns for a while... until they completely destroy it. A famous example is the number of regions you can divide a circle by n chords. It goes 1, 2, 4, 8, 16... 31. Why should 5 have suddenly decided it didn't want to divide a circle into 32 regions? You might argue that the sixth number of a pattern is too small to compare to primes, but then what is that magic number? 10? 100? 1 million? 1 googolplex? Grahm's number? Numbers merely have properties prescribed to them by rigid formulae. Anthropomorphizing them as to deciding whether or not they can choose to have a value is pointless: their constructions dictate that they must have a certain property, as explained by a formula, or not. -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth |